# Coulomb interaction

Written by Yuhang Jin, AP225 2011 Fall.

## Introduction

Coulomb interaction is the electrostatic interactions between electric charges, and follows the Coulomb's law, which is a basis of classical electrodynamics. In general, Coulomb interaction can manifest itself on various scales from microscopic particles to macroscopic bodies. The microscopic theory of Coulomb interaction has been developed in the frame of quantum field theory.

## Basic properties and formulae of Coulomb interaction

The electrostatic interaction of two point charges $q_1$ at position $\mathbf r_1$ and $q_2$ at position $\mathbf r_2$ is described by the Coulomb's law:

$\mathbf F_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}$,

where $\mathbf F$ is the electrostatic force experience by $q_1$ (solely) due to the presence of $q_2$, and $\epsilon_0$ is the vacuum permittivity, which has a value of $8.85\times10^{-12}$ approximately.

Coulomb interaction obeys the principle of linear superstition, and therefore the force on a point charge at position $\mathbf r$ exerted by a object with continuous charge distribution is give as

$\mathbf F=\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r')(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}d^3\mathbf r'$,

where $\rho(\mathbf r')$ is the charge density of the body, and furthermore, the Coulomb interaction between two bodies with continuous charge distribution is

$\mathbf F=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)(\mathbf r_1-\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|^3}d^3\mathbf r_1d^3\mathbf r_2$.

Coulomb forces are conservative, and as a result Coulomb interaction can be described using the Coulomb potential. The potential created by a point charge $q_1$ is

$U=\frac{1}{4\pi\varepsilon_0}\frac{q_1}{r}$,

where $r$ is the distance from $q_1$ charge. The force experienced by another point charge $q_2$ due to $q_1$ is given by

$\mathbf F_{21}=-q_2\nabla U$.

The interaction potential between two point charges is given as

$U_{12}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{|\mathbf r_1-\mathbf r_2|}$.

Similarly, the Coulomb interaction potential of two bodies with continuous charge distribution is

$U_{12}=\int\int\frac{1}{4\pi\varepsilon_0}\frac{\rho(\mathbf r_1)\rho(\mathbf r_2)}{|\mathbf r_1-\mathbf r_2|}d^3\mathbf r_1d^3\mathbf r_2$,

## Applications of Coulomb interaction

Almost all the aspects of electrical engineering rely more or less on the theory of Coulomb interaction. Coulomb interaction is also the key issue various physical processes, including the formation of electrical double layers and the stabilization of colloids.

## References

[1] Jackson, J. D., "Classical Electrodynamics", John Wiley & Sons, 1999.